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arXiv:1002.1817

A mechanism for pair formation in strongly correlated systems

Tobias G. W. Verhulst, Jan Naudts

Published Feb 9, 2010• 0 citations
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Abstract

We start from a Hamiltonian describing non-interacting fermions and add bosons to the model, with a Jaynes-Cummings-like interaction between the bosons and fermions. Because of the specific form of the interaction the model can be solved exactly. In the ground state, part of the electrons form bound pairs with opposite momentum and spin. The model also shows a gap in the kinetic energy of the fermions, but not in the spectrum of the full Hamiltonian. This gap is not of a mean-field nature, but is due to the Pauli exclusion principle.

DOI
10.48550/arxiv.1002.1817

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We start from a Hamiltonian describing non-interacting fermions and add bosons to the model, with a Jaynes-Cummings-like interaction between the bosons and fermions. Because of the specific form of the interaction the model can be solved exactly. In the ground state, part of the electrons form bound pairs with opposite momentum and spin. The model also shows a gap in the kinetic energy of the fermions, but not in the spectrum of the full Hamiltonian. This gap is not of a mean-field nature, but is due to the Pauli exclusion principle.
The boson-fermion model, describing a mixture of itinerant electrons hybridizing with tightly bound electron pairs represented as hardcore bosons, is here generalized with the inclusion of a term describing on-site Coulomb repulsion between fermions with opposite spins. Within the general framework of the dynamical mean-field theory, it is shown that around the symmetric limit of the model this interaction strongly competes with the local boson-fermion exchange mechanism, smoothly driving the system from a pseudogap phase with poor conducting properties to a metallic regime characterized by a substantial reduction of the fermionic density. On the other hand, if one starts from correlated fermions described in terms of the one-band Hubbard model, the introduction in the half-filled insulating phase of a coupling with hardcore bosons leads to the disappearance of the correlation gap, with a consequent smooth crossover to a metallic state.
While the one-Cooper-pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for N=2N=2N=2 already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to NNN pairs but no precise description of this continuous change has been given so far. Using Richardson's procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs---the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.
In cuprate superconductors, superconductivity is consistently accompanied by antiferromagnetism. This raises the question of a potential causal link between superconductivity and antiferromagnetic mechanisms. In this study, we consider the non-local Coulomb interaction and solve the single-particle Green function exactly. The exact solution shows the existence of strong inter-spin correlations in the absence of superexchange, eliminating the directional degrees of freedom for spin quantization in the Hatsugai-Kohmoto (HK) model. Strong Coulomb interactions force the energy band splitting into infinitely equidistant sub-bands weighted by a Poisson distribution, exhibiting boson properties (incoherent pairs of fermions) and non-Fermi liquids. Attractive pairing interactions caused by electron-phonon coupling open a pairing gap at the Fermi surface. Based on the Pauli exclusion principle, paired particles with the same spin quantization direction completely occupy the state below the gap, resulting in pairing insulators, superconducting instability, and especially antiferromagnetism. A pseudogap occurs when the higher split sub-band overlaps in energy with the lowest sub-band. The critical temperature Tc of superconductivity exhibits a multi-dome structure and peaks around the maximum density of states. The maximum Tc depends on splitting sub-bands and particle occupancy, consistent with experiments. This study concludes that the non-local effect preserves the Hubbard and HK models' characteristics while renormalizing the system to antiferromagnetism in pairing insulators and Pauli paramagnetism in weak semimetals.